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Mathematical property of subsets in order theory
of a preordered set ( A , ≤ ) {\displaystyle (A,\leq )} is said to be cofinal or frequent in A {\displaystyle A} if for every a ∈ A , {\displaystyle
Cofinal_(mathematics)
Size of subsets in order theory
especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. Formally, cf
Cofinality
System of pitch organization in Gregorian chant
below, C, remained the lower limit. In addition to the range, the tenor (cofinal, or dominant, corresponding to the "reciting tone" of the psalm tones)
Gregorian_mode
Topics referred to by the same term
Cofinal may refer to: Cofinal (mathematics), the property of a subset B of a preordered set A such that for every element of A there is a "larger element"
Cofinal
Generalization of "n-th" to infinite cases
The cofinality of any ordinal α is a regular ordinal, i.e. the cofinality of the cofinality of α is the same as the cofinality of α. So the cofinality operation
Ordinal_number
Generalization of the concept of subsequence to the case of nets
h ( I ) {\displaystyle h(I)} is cofinal in A . {\displaystyle A.} The set h ( I ) {\displaystyle h(I)} being cofinal in A {\displaystyle A} means that
Subnet_(mathematics)
Infinite Cardinal number
indexed by ℶ {\displaystyle \beth } . On the other hand, beth numbers are cofinal (every cardinal number is less than a beth number) in plain Zermelo-Fraenkel
Beth_number
Well-quasi-ordering of finite trees
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Kruskal's_tree_theorem
Field in mathematics similar to the real numbers
Archimedean property is related to the concept of cofinality. A set X contained in an ordered set F is cofinal in F if for every y in F there is an x in X such
Real_closed_field
Subset of a preorder that contains all larger elements
isomorphism) in this way as the lattice of lower sets of a unique finite poset. Cofinal set – a subset U {\displaystyle U} of a partially ordered set ( X , ≤ )
Upper_and_lower_sets
Generalization of a category
a final map. Also, a map f : X → Y {\displaystyle f:X\to Y} is called cofinal if f : X o p → Y o p {\displaystyle f:X^{op}\to Y^{op}} is final. Presheaf
Quasi-category
Infinite cardinal number
cardinals with cofinality ℵ 0 {\displaystyle \aleph _{0}} . An uncountably infinite cardinal κ {\displaystyle \kappa } having cofinality ℵ 0 {\displaystyle
Aleph_number
Mathematical result on order relations
consequence of the axiom of choice, the principle that every total order has a cofinal well-order, can be combined to prove the full axiom of choice. With these
Szpilrajn_extension_theorem
Order-preserving mathematical function
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Monotonic_function
Four mathematical theorems
D {\displaystyle C\subset D} be exact categories. Then C is said to be cofinal in D if (i) it is closed under extension in D and if (ii) for each object
Basic theorems in algebraic K-theory
Basic_theorems_in_algebraic_K-theory
Class of mathematical orderings
the whole set. Subsets that are unbounded in the whole set. A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it
Well-order
Generalization of a sequence of points
x_{\bullet }=\left(x_{a}\right)_{a\in A}} is said to be frequently or cofinally in S {\displaystyle S} if for every a ∈ A {\displaystyle a\in A} there
Net_(mathematics)
mathematical theory, introduced by Saharon Shelah (1978), that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on
Pcf_theory
Extreme element of a preorder
{\displaystyle Q} of a partially ordered set P {\displaystyle P} is said to be cofinal if for every x ∈ P {\displaystyle x\in P} there exists some y ∈ Q {\displaystyle
Maximal_and_minimal_elements
Type of large transfinite number
an initial subsequence of the cf(κ)-sequence. Thus its cofinality is less than the cofinality of κ and greater than it at the same time; which is a contradiction
Mahlo_cardinal
Spanish bus driver (1888–1959)
female bus driver. She went on to establish the first bus line between Cofiñal and Boñar, which she had initially established in 1908 using a horse-drawn
Catalina_García_González
theory, the notion of final functor is a generalization of the notion of cofinal set from order theory. A functor F : C → D {\displaystyle F:C\to D} is
Final_functor
German mathematician (1868–1942)
-1}^{\aleph _{\alpha }}.} This formula was, together with a later notion called cofinality introduced by Hausdorff, the basis for all further results for Aleph exponentiation
Felix_Hausdorff
Type of monotone function
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Order_embedding
Collection of mathematical objects of finite size
Euclidean distance. A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater
Bounded_set
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Type of ordering of a set
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Dense_order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Prefix_order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Series-parallel_partial_order
Set theory concept
following statement: 2cf(κ) < κ implies κcf(κ) = κ+, where cf denotes the cofinality function. Note that κcf(κ)= 2κ for all singular strong limit cardinals
Singular_cardinals_hypothesis
On chains and antichains in partial orders
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Dilworth's_theorem
Mathematical object
E_{j}} suspends to an (i + 1)-cell in E j + 1 {\displaystyle E_{j+1}} , a cofinal subspectrum is a subspectrum for which each cell of the parent spectrum
Spectrum_(topology)
Ideals in a Boolean algebra can be extended to prime ideals
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Boolean_prime_ideal_theorem
Proposition in mathematical logic
disproof and established Kőnig's theorem, which by using the concept of cofinality introduced in 1908 by Felix Hausdorff, shows that result that 2 ℵ 0 {\displaystyle
Continuum_hypothesis
Set theory concept
set). Let κ {\displaystyle \kappa \,} be a limit ordinal of uncountable cofinality λ . {\displaystyle \lambda .} For some α < λ {\displaystyle \alpha <\lambda
Club_set
Algebraic object with an ordered structure
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Ordered_field
Glossary of terms used in branch of mathematics
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Glossary_of_order_theory
Mathematical theorem in set theory
{\displaystyle \kappa <\operatorname {cf} (2^{\kappa })} (where cf(α) is the cofinality of α) and if κ < λ then 2 κ ≤ 2 λ . {\displaystyle {\text{if }}\kappa
Easton's_theorem
Existence of certain infima or suprema of a given poset
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Completeness_(order_theory)
Large cardinal property in set theory
into itself then α {\displaystyle \alpha } is either a limit ordinal of cofinality ω {\displaystyle \omega } or the successor of such an ordinal. The axioms
Rank-into-rank
Mathematical result or axiom on order relations
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Hausdorff_maximal_principle
Type of logical relation
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Total_relation
Alternative mathematical ordering
quotient L / Z, where L is a linearly ordered group and Z is a cyclic cofinal subgroup of L. Every cyclically ordered group can also be expressed as
Cyclic_order
Mathematical operation
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Composition_of_relations
Set whose pairs have minima and maxima
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Lattice_(order)
Subset of incomparable elements
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Antichain
Equivalence of partially ordered sets
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Order_isomorphism
Order whose elements are all comparable
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Total_order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
List of order structures in mathematics
List_of_order_structures_in_mathematics
Special subset of a partially ordered set
case where μ is counting measure. Given an ordinal a with uncountable cofinality, a subset of a is called a club if it is closed in the order topology
Filter_(mathematics)
Vector space with a partial order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Ordered_vector_space
Mathematical property of algebraic structures
Archimedean fields in terms of these substructures. The natural numbers are cofinal in K {\displaystyle K} . That is, every element of K {\displaystyle K}
Archimedean_property
Reversal of the order of elements of a binary relation
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Converse_relation
superior and limit inferior Irreducible element Prime element Compact element Cofinal and coinitial set, sometimes also called dense Meet-dense set and join-dense
List_of_order_theory_topics
Generalization of the real numbers
class of ordinal numbers, and because O n {\textstyle \mathbb {On} } is cofinal in N o {\textstyle \mathbb {No} } we have { N o ∣ } = { O n ∣ } = O n {\textstyle
Surreal_number
Mathematical ordering of a partial order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Linear_extension
Class of cardinal numbers
\ldots \}=\bigcup _{n<\omega }\beth _{n}} is a strong limit cardinal of cofinality ω. More generally, given any ordinal α, the cardinal ℶ α + ω = ⋃ n < ω
Limit_cardinal
There are equally many countable order types and real numbers
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Cantor–Bernstein_theorem
Construction in order theory
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Star_product
Relationship between elements of two sets
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Binary_relation
Mathematical ordering with upper bounds
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Directed_set
Partially ordered set in which all subsets have both a supremum and infimum
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Complete_lattice
Sequence of points that get progressively closer to each other
multiples of p r . {\displaystyle p_{r}.} If H {\displaystyle H} is a cofinal sequence (that is, any normal subgroup of finite index contains some H
Cauchy_sequence
American mathematician
MS16. Malliaris, M.; Shelah, S. (2016), "Cofinality spectrum theorems in model theory, set theory, and general topology", Journal of the American Mathematical
Maryanthe_Malliaris
Type of topology in mathematics
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Alexandrov_topology
Bound lattice in which every element has a complement
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Complemented_lattice
Isomorphism type of ordered sets
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Order_type
Mathematical set with an ordering
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Partially_ordered_set
Certain topology in mathematics
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Order_topology
Property of elements related by inequalities
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Comparability
Partially ordered vector space, ordered as a lattice
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Riesz_space
Nonempty, upper-bounded, downward-closed subset
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Ideal_(order_theory)
Basic integral in elementary calculus
all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions. Another popular restriction is the
Riemann_integral
Partially ordered set equipped with a rank function
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Graded_poset
Large cardinal number
not equivalent; in fact, the smallest worldly cardinal has countable cofinality and therefore is a singular cardinal. The following are in strictly increasing
Worldly_cardinal
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Better-quasi-ordering
Concept in mathematics
{\displaystyle x} if and only if B {\displaystyle {\mathcal {B}}} is a cofinal subset of ( N ( x ) , ⊇ ) {\displaystyle \left({\mathcal {N}}(x),\supseteq
Neighbourhood_system
Construction for simplicial sets
}\times A)\times _{B^{\mathrm {op} }\times B}\operatorname {Tw} (B),} are cofinal. Cisinski, Denis-Charles (2019-06-30). Higher Categories and Homotopical
Twisted diagonal (simplicial sets)
Twisted_diagonal_(simplicial_sets)
Characterizes the height of any finite partially ordered set
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Mirsky's_theorem
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Symmetric_closure
Visual depiction of a partially ordered set
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Hasse_diagram
Mathematical relation inside orderings
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Covering_relation
This implies that V and V[G] have the same cardinals (and the same cofinalities). A subset D of P is called dense if for every p ∈ P there is some q
List_of_forcing_notions
Banach space with a compatible structure of a lattice
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Banach_lattice
Theorem in axiomatic set theory
\kappa \mapsto \kappa ^{\mathrm {cf} (\kappa )}} where cf denotes the cofinality function; the gimel function is used for studying the continuum function
Gimel_function
Term in the mathematical area of order theory
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Duality_(order_theory)
Mathematical proposition equivalent to the axiom of choice
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Zorn's_lemma
Graph linking pairs of comparable elements in a partial order
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Comparability_graph
Axiom in the mathematical field of set theory
dense subsets. If P is a non-empty upwards ccc poset and Y is a set of cofinal subsets of P with |Y| ≤ κ then there is an upwards-directed set A such
Martin's_axiom
Non-empty family of sets that is closed under finite unions and subsets
numbers. If λ {\displaystyle \lambda } is an ordinal number of uncountable cofinality, the nonstationary ideal on λ {\displaystyle \lambda } is the collection
Ideal_on_a_set
Mathematical function on ordinals
_{\alpha }(\beta )<\delta } ). The fundamental sequence for an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence that has the ordinal
Veblen_function
Concept in set theory
_{2}} to an ordinal of cofinality ω {\displaystyle \omega } . Let G {\displaystyle G} be an ω {\displaystyle \omega } -sequence cofinal on ω 2 L {\displaystyle
Zero_sharp
Set-theoretic concept
a powerset. If κ {\displaystyle \kappa } is a cardinal of uncountable cofinality, S ⊆ κ , {\displaystyle S\subseteq \kappa ,} and S {\displaystyle S} intersects
Stationary_set
Theorem in set theory
the theorem. Kőnig's theorem has also important consequences for the cofinality of cardinal numbers. If κ ≥ ℵ 0 {\displaystyle \kappa \geq \aleph _{0}}
Kőnig's_theorem_(set_theory)
Cardinality of the set of real numbers
cases, equality can be ruled out by König's theorem on the grounds of cofinality (e.g. c ≠ ℵ ω {\displaystyle {\mathfrak {c}}\neq \aleph _{\omega }} )
Cardinality_of_the_continuum
vector lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet
Reflexive_closure
Set theory concept
cardinal characteristics. Stephen Hechler. On the existence of certain cofinal subsets of ω ω {\displaystyle {}^{\omega }\omega } . In T. Jech (ed), Axiomatic
Cardinal characteristic of the continuum
Cardinal_characteristic_of_the_continuum
Mathematical concept
closed and unbounded subset of κ, so that for every λ in C of uncountable cofinality, there is an unbounded subset of λ that is homogenous for f; slightly
Ramsey_cardinal
I)(\exists A\in {\mathcal {A}})(B\subseteq A){\big \}}.} The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must
Cichoń's_diagram
American mathematician (1942–2016)
Problem", Silver proved that if a cardinal κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Prior to Silver's
Jack_Silver
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Biblical
a scab
Surname or Lastname
English
English : variant of Stratford.
Boy/Male
British, English
Strong Guardian
Boy/Male
Bengali, Hindu, Indian
Sun Ray
Boy/Male
English
Son of a dark man.
Boy/Male
Muslim
Flower name
Boy/Male
Indian, Sanskrit, Telugu
Sign
Boy/Male
Hindu
Boy/Male
Indian, Punjabi, Sikh
Brave and Joyous
Boy/Male
Arabic, Muslim
Accept; Submission
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